T(x1,x2,x3,x4) = (4x1-x2, 5x2 + 3x4, -x3)

Image = (4,0,0) (-1,5,0) (0,0,-1) (0,3,0)

So the vectors that form a basis are (4,0,0) (0,0,-1) (0,3,0)

Kernel

4x1 - x2 = 0

5x2 + 3x4 = 0

-x3 = 0

x1 = 1/4 x2

x2 = x2

x3 = 0

x4 = -5/3 x2

(x1, x2, x3, x4) = (1/4x2, x2, 0, -5/3 x2)

= x2(1/4, 1, 0, -5/3)

Kernel = (1/4, 1, 0, -5/3)